101445: [AtCoder]ABC144 F - Fork in the Road

Problem Statement

There is a cave consisting of $N$ rooms and $M$ one-directional passages. The rooms are numbered $1$ through $N$.

Takahashi is now in Room $1$, and Room $N$ has the exit. The $i$-th passage connects Room $s_i$ and Room $t_i$ ($s_i$ < $t_i$) and can only be traversed in the direction from Room $s_i$ to Room $t_i$. It is known that, for each room except Room $N$, there is at least one passage going from that room.

Takahashi will escape from the cave. Each time he reaches a room (assume that he has reached Room $1$ at the beginning), he will choose a passage uniformly at random from the ones going from that room and take that passage.

Aoki, a friend of Takahashi's, can block one of the passages (or do nothing) before Takahashi leaves Room $1$. However, it is not allowed to block a passage so that Takahashi is potentially unable to reach Room $N$.

Let $E$ be the expected number of passages Takahashi takes before he reaches Room $N$. Find the value of $E$ when Aoki makes a choice that minimizes $E$.

Constraints

• $2 \leq N \leq 600$
• $N-1 \leq M \leq \frac{N(N-1)}{2}$
• $s_i < t_i$
• If $i != j$, $(s_i, t_i) \neq (s_j, t_j)$. (Added 21:23 JST)
• For every $v = 1, 2, ..., N-1$, there exists $i$ such that $v = s_i$.

Input

Input is given from Standard Input in the following format:

$N$ $M$
$s_1$ $t_1$
$:$
$s_M$ $t_M$

Output

Print the value of $E$ when Aoki makes a choice that minimizes $E$. Your output will be judged as correct when the absolute or relative error from the judge's output is at most $10^{-6}$.

Sample Input 1

4 6
1 4
2 3
1 3
1 2
3 4
2 4

Sample Output 1

1.5000000000

If Aoki blocks the passage from Room $1$ to Room $2$, Takahashi will go along the path 1 → 3 → 4 with probability $\frac{1}{2}$ and 1 → 4 with probability $\frac{1}{2}$. $E = 1.5$ here, and this is the minimum possible value of $E$.

Sample Input 2

3 2
1 2
2 3

Sample Output 2

2.0000000000

Blocking any one passage makes Takahashi unable to reach Room $N$, so Aoki cannot block a passage.

Sample Input 3

10 33
3 7
5 10
8 9
1 10
4 6
2 5
1 7
6 10
1 4
1 3
8 10
1 5
2 6
6 9
5 6
5 8
3 6
4 8
2 7
2 9
6 7
1 2
5 9
6 8
9 10
3 9
7 8
4 5
2 10
5 7
3 5
4 7
4 9

Sample Output 3

3.0133333333